Two sample t test  equal variances not assumed  overview
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Two sample $t$ test  equal variances not assumed  $z$ test for the difference between two proportions  Marginal Homogeneity test / StuartMaxwell test  Sign test 


Independent/grouping variable  Independent/grouping variable  Independent variable  Independent variable  
One categorical with 2 independent groups  One categorical with 2 independent groups  2 paired groups  2 paired groups  
Dependent variable  Dependent variable  Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  One categorical with $J$ independent groups ($J \geqslant 2$)  One of ordinal level  
Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  
H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.  H_{0}: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2.  H_{0}: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$ 
 
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\mu_1 \neq \mu_2$ H_{1} right sided: $\mu_1 > \mu_2$ H_{1} left sided: $\mu_1 < \mu_2$  H_{1} two sided: $\pi_1 \neq \pi_2$ H_{1} right sided: $\pi_1 > \pi_2$ H_{1} left sided: $\pi_1 < \pi_2$  H_{1}: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group. 
 
Assumptions  Assumptions  Assumptions  Assumptions  



 
Test statistic  Test statistic  Test statistic  Test statistic  
$t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1  \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2  \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.  $z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2  p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$  Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.  $W = $ number of difference scores that is larger than 0  
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  Sampling distribution of the test statistic if H_{0} were true  Sampling distribution of $W$ if H_{0} were true  
Approximately the $t$ distribution with $k$ degrees of freedom, with $k$ equal to $k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1  1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2  1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2}$ or $k$ = the smaller of $n_1$  1 and $n_2$  1 First definition of $k$ is used by computer programs, second definition is often used for hand calculations.  Approximately the standard normal distribution  Approximately the chisquared distribution with $J  1$ degrees of freedom  The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$.
If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1P)} = \sqrt{n \times 0.5(1  0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W  n \times 0.5}{\sqrt{n \times 0.5(1  0.5)}}$$ follows approximately the standard normal distribution if the null hypothesis were true.  
Significant?  Significant?  Significant?  Significant?  
Two sided:
 Two sided:
 If we denote the test statistic as $X^2$:
 If $n$ is small, the table for the binomial distribution should be used: Two sided:
If $n$ is large, the table for standard normal probabilities can be used: Two sided:
 
Approximate $C\%$ confidence interval for $\mu_1  \mu_2$  Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  n.a.  n.a.  
$(\bar{y}_1  \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$
where the critical value $t^*$ is the value under the $t_{k}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu_1  \mu_2$ can also be used as significance test.  Regular (large sample):
     
Visual representation  n.a.  n.a.  n.a.  
      
n.a.  Equivalent to  n.a.  Equivalent to  
  When testing two sided: chisquared test for the relationship between two categorical variables, where both categorical variables have 2 levels.   
Two sided sign test is equivalent to
 
Example context  Example context  Example context  Example context  
Is the average mental health score different between men and women?  Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?  Do people tend to score higher on mental health after a mindfulness course?  
SPSS  SPSS  SPSS  SPSS  
Analyze > Compare Means > IndependentSamples T Test...
 SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
Jamovi  Jamovi  n.a.  Jamovi  
TTests > Independent Samples TTest
 Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples  $\chi^2$ test of association
   Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:
ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  Practice questions  Practice questions  